Theorem 3.1 (Hardt–Simon [1]}) 
If \(\mathcal {C}^n \subset \mathbb {R}^{n+1}\)  is a regular area minimizing cone then there exists smooth area-minimizing hypersurfaces \(S_{\pm }\)  in each component of \(\mathbb {R}^{n+1}\setminus \mathcal {C}= U_+ \cup U_-\)  so that if \(S^{\prime }\)  is area minimizing and contained in \(U_\pm \)  then \(S^{\prime } = \lambda S_{\pm }\) .

